$6^{+}$ isomers in neutron-rich Sn isotopes beyond $N=82$ and effective interactions

Recent observation of the $6^{+}$ seniority isomers and measurements of the $B\left(E2\right)$ values in the ${}^{134\text{–}138}\mathrm{Sn}$ isotopes lying close to the neutron drip line have raised some questions about the validity of the currently used effective interactions in the neutron-rich region. Simpson et al. [Phys. Rev. Lett. 113, 132502 (2014)] had to modify the diagonal and nondiagonal $\nu f_{7/2}^2$ two-body matrix elements of the $Vlk$ interaction by $\sim 150$ keV in their shell model calculations in order to explain the data of ${}^{136}\mathrm{Sn}$. In contrast, we are able to explain the observed energy levels and the $B\left(E2\right)$ values after marginal reduction of the same set of matrix elements by $25$ keV in the RCDB (renormalized CD-Bonn) interaction. The observed mismatch in reproducing the data of ${}^{136}\mathrm{Sn}$ is due to the seniority mixing. Further, we do not find it necessary to consider the core excitations, and the RCDB interaction seems better suited to explain the data beyond $N=82$ magic number.

Figure 1

Energy level schemes for ${}^{134\text{–}138}\mathrm{Sn}$. The experimental data shown on the left are taken from Refs. [2, 4, 5, 6] and results of the shell model calculations with RCDB interaction and a modified RCDB (RCDBMO) interaction (see text for details) are shown on the right. The second $4_2^{+}$ state has been shown only for ${}^{136,138}\mathrm{Sn}$ and lies above the yrast $4_1^{+}$ state. All the energies are in keV. The $B\left(E2\right)$ values in units of ${\text{e}}^2{\text{fm}}^4$, shown in $\mathit{italics}$, have been rounded off.

Figure 2

Evolution of effective single-particle energies of different orbitals with increasing neutron number for the RCDB interaction. The $f_{7/2}$ orbital is also plotted for the RCDBMO interaction. ${}^{132}\mathrm{Sn}$ is taken as the core.

Figure 3

Reduced transition rates for the yrast $6^{+}\to 4_1^{+}$ transitions in ${}^{134\text{–}138}\mathrm{Sn}$. Experimental data and error bars are taken from Refs. [2, 4]. Calculated values for both RCDB and RCDBMO interactions are also shown.

Figure 4

Average occupancy (in percentage) of the $f_{7/2}$ and $p_{3/2}$ orbitals for the $4_1^{+},4_2^{+}$, and $6^{+}$ states in ${}^{134\text{–}138}\mathrm{Sn}$ isotopes.